I always prefer the Hero's Bow over the Slingshot without a doubt. Maybe that's because I tend to enjoy picking off targets (such as the Bulblin Archers) from a long range. If you're "sniping" with the Slingshot, you have to take projectile motion physics into account, which requires a lot of trial and error for me; arrows, though, travel straight, so it's far better to use the Bow for long range.
Another thing I really like about the Bow over the Slingshot is that finding ammo is easy. When I run out of arrows, it's never a problem because I know arrows are everywhere. I rarely ever see pumpkin seed refills though.
And, of course, arrows are more powerful and hurt more enemies than pumpkin seeds.